Existence and uniqueness of wave solutions have been investigated for decades. Recent analytical and computational efforts are focused on the behavior of the wave solutions. In the present work, a nonlocal hyperbolic-parabolic population model with maturation and dispersal delays is analyzed. The model exhibits a delay dependent Hopf bifurcation threshold, which influences the spatio-temporal patterns of population densities. Further analysis of the model indicates the loss of stability of the stationary wave solutions due to increased values of dispersal delay. The population models with maturation and dispersal delays can give new insights into the complex dynamics of invasive species.

Gene regulatory networks (GRNs), frequently modeled using Boolean networks, describe how a collection of genes governs the processes within a cell. Boolean networks are intuitive, simple to describe, and yield qualitative results even given sparse data. The biological term canalization reflects a cell’s ability to maintain a stable phenotype despite ongoing environmental perturbations. Accordingly, Boolean canalizing functions are functions where the output is already determined if a certain, canalizing variable takes on its canalizing input, regardless of all other inputs. Due to their biological meaningfulness, this class of functions has been hypothesized to be overrepresented in GRNs and used as an explanation for the observed stability of GRNs; however, published GRNs have never been thoroughly evaluated to test this hypothesis. Using text- and data-mining techniques and the PubMed search engine, we generated an expandable database of published Boolean GRNs, and extracted the rules governing these networks. A meta-analysis of all extracted curated rules confirmed a strong overrepresentation of certain types of canalizing functions. We further studied the relationship between network topology, stability and limit behavior and found significant differences between the published GRNs but also when comparing to random networks. These findings highlight how our continuously-expanding database provides a versatile tool for various meta-analyses.

This seminar presents a model for Neisseria meningitidis and the application of optimal and impulse controls to investigate the best strategy for curtailing the disease in Nigeria. The control strategies uniquely incorporate the use of an impulsive-like optimal facial mask usage during a specific time frame. A comparison of the infection averted using the two controls shows the advantage of using facial mask as more infection are averted unlike in the absence of it.

Anti-predator behavior of prey due to fear of predators is a well observed phenomenon in natural predator-prey communities. However, such dynamics have not been well investigated mathematically. In the present work, we investigate the effect of fear induced defense in a predator-prey system. We generalize defense in this paper to include mimicry, group or herd defense, apostatic selection, playing dead, camouflage and toxin production. We investigate mathematical properties such as positivity, ultimate boundedness, dissipativeness, uniform persistence, local and global stability analysis, and bifurcation analysis of the model subjected to Holling type-I and type-II functional responses. Moreover, we speculate various ecological outcomes via of optimal control analysis. We perform extensive numerical experiments to visualize the dynamical behavior of the system to correlate our analytical findings.

The Trojan Y-Chromosome (TYC) strategy is a biological control used to cause the extinction of invasive sh populations by manipulating the populations sex ratio. This is done by introducing YY super males into the invasive population, which guarantees male progeny upon mating with the YY males. This leads to a decline of wild type females in the population and eventually leads to extinction of the population. In this work, we show that, if the introduction rate of the trojan-sh is zero under certain large data assumptions, it is possible to obtain negative solutions for the male population which in turn can lead to a nite-time blow up in the female population. We support our work with numerical solutions.

Evolution often plays a fundamental role in determining the persistence of natural populations, including those beneficial to us, such as keystone species, or harmful, such as pathogens and invasive species. Hence, understanding how populations evolve has the potential to shape how we think about conservation as well as combating diseases. One of the classical ways we've thought populations evolve is through continual and novel mutations, some of which may be beneficial and allow a population to adapt to new environments. More recently, scientists have suggested that certain proteins that mask the function of mutations, may allow for more mutations to accumulate and be "stored", for when they are needed. These proteins are referred to as "evolutionary capacitors". In this talk, I will give examples of proteins hypothesized to do this and then explore a general model of how epistatic (nonlinear) interactions between proteins and genetic mutations may allow many proteins to play this role.

Following injury, skin activates a complex wound healing program. While cellular and signaling mechanisms of wound repair have been extensively studied, the principles of epidermal‐dermal interactions and their effects on wound healing outcomes are only partially understood. To gain new insight into the effects of epidermal‐dermal interactions, we developed a multiscale, hybrid mathematical model of skin wound healing. The model takes into consideration interactions between epidermis and dermis across the basement membrane via diffusible signals, defined as activator and inhibitor. Simulations revealed that epidermal‐dermal interactions are critical for proper extracellular matrix deposition in the dermis, suggesting these signals may influence how wound scars form. Our model makes several theoretical predictions. Taken together, our model predicts the important role of signaling across dermal‐epidermal interface and the effect of fibrin clot density and wound geometry on scar formation. This hybrid modelling approach may be also applicable to other complex tissue systems, enabling the simulation of dynamic processes, otherwise computationally prohibitive with fully discrete models due to a large number of variables.

Languages compete, just like species do, for speakers in a population for social and economically advantageous. Thousands of world's languages are vanishing at an alarming rate and over half the worlds languages are predicted to go extinct in this century alone. We survey current language dynamics models and discuss new models of language extinction. We then apply these new models to historical data on current endangered languages as well as extinct languages to describe their decline and death respectively. We hope to inform language protection policy.